## 2. The steady equilibrium stateWe model a sharp interface by two semi-infinite homogeneous regions
separated by a nonuniform layer. In the Cartesian coordinate system we
use, the
The magnetic field is assumed to be homogeneous throughout the whole space whereas the plasma flow, with velocity parallel to the magnetic field, exists only for . In particular : Thus the flow speed is discontinuous at , while the other physical quantities such as density and temperature are assumed to behave smoothly through the nonuniform layer. In our treatment we ignore gravity, so that the magnetohydrostatic equation is simply given by This means that the thermal pressure is uniform and the plasma parameter constant where while and are the squares of the Alfvén and the sound speed respectively. The Alfvén speed is assumed to have a linear profile through the nonuniform layer and to be constant elsewhere : Since is constant (2), the speed of sound is simply proportional to : We also introduce the cusp speed which is defined as Besides the plasma parameter, the configuration is also characterized by the temperature ratio , which can be written in terms of the Alfvén speeds in the two homogeneous regions as According to the definition of the Alfvén speed and from (3) and (4) the density profile in the nonuniform layer takes the following form : For the rest of the paper, length, speed, density and magnetic
field strength are non-dimensionalized with respect to
, ,
and respectively. and
are the The unperturbed plasma is taken to be ideal because the dissipative effects can be neglected on the time scale of the MHD wave propagation. © European Southern Observatory (ESO) 1998 Online publication: March 23, 1998 |